# Constraint Satisfaction Problem: who killed Warren White?

Warren White was found shot dead one morning, police have then investigated who was behind the incident. There were 3 suspects who could be suspects in the incident. That afternoon, the 3 suspects stated that:

• Billy:

1. I didn't do it
2. I've never seen Jerry before
3. It's true, that I know Warren
• Jerry:

1. I didn't do it
2. Billy and Tom are my friends
3. Billy didn't kill anyone
• Tom:

1. I didn't do it
2. Billy lied when he said he had never seen Jerry before
3. I don't know who did it

If only one of the three statements above is false, and only one of the three people is guilty, who is Warren's killer?

• When you say "one of the three statements", do we consider the three bullet points from each suspect as one statement? Jun 16 at 9:49

TOM is Warren's killer

Billy's second remark

I've never seen Jerry before

and Tom's second remark

Billy lied when he said he had never seen Jerry before

Are contradictory, therefore one of them gave the false statement. Since only one statement is false,
Jerry's statement is true, and his remark

Billy didn't kill anyone

is true; therefore Billy and Jerry's remarks

I didn't do it

Are true, and TOM is Warren's killer.
Since Tom said

I didn't do it

Tom is the liar, so Billy's remarks

I've never seen Jerry before
It's true, that I know Warren

are true. What do we make of Jerry's remark "Billy and Tom are my friends",
which must be true, although Billy has never seen Jerry?
Perhaps Billy is blind, or perhaps they are pen-pals.

1) Tom is innocent. He made only one false statement, so Tom1, Tom3, or both must be true. In any case, he is innocent.
2) Since one of the three is guilty, either Jerry1 or Jerry3 is false. In either case, Jerry2 is true.
3) Jerry2 and Billy2 contradict each other. Since Jerry2 is true, Billy2 is false. Since Billy2 is false, Billy1 is true; Billy is innocent.
4) Since Billy and Tom are innocent, Jerry is guilty.
5) Since Jerry is guilty, Jerry1 is false.
6) Since Billy2 is false, Tom2 is true. Since Tom is innocent, Tom1 is true. By elimination, Tom3 is false.

Final answer: Jerry is guilty; false statements are Billy2, Jerry1, and Tom3.

Billy 2nd and Jerry 2nd is contradictory. So one of them must be false.

If Billy 2nd is false, then Billy 1st and 3rd are true. Then Jerry 3rd is also true. Which means Jerry 1st is false (So Jerry is the killer)

Tom 1st and 2nd is true, so Tom 3rd is false.

The first statement for all three is "I didn't do it". It must be false for the killer and true for the two innocent men. So, killer's statement 2 and 3 must be both true. For those who are innocent, either statement 2 or statement 3 must be false.

Billy's statement 2 and Jerry and Tom's statement 2 are contradictory. Suppose Billy's statement 2 (that he's never met Jerry) is true, then Jerry and Tom's statement 2 must both be false. In that case, both are innocent (since statement one must be true for both of them). In that case, Billy must be the killer. But then, Jerry's statement 3 must be false. In that case both Jerry's statement 2 and 3 would both be false, which contradicts the constraints. So, we know that Billy is innocent, and his statement 2 is false, so both Jerry and Tom's statement 2 is true. Therefore, the killer is either Tom or Jerry.

Suppose Tom is the killer. In that case, his statement 1 and 3 would both be false (if he's the killer, he also has to know who the killer is - it's him). That violates the constraints, so Tom is innocent.

That leaves Jerry as the killer. Constraints check out, his statement 2 is true, nothing proves that his statement 3 is false, so his statement 1 may be false.